Differential Equations (Mathematics)

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1 H I SHIVAJI UNIVERSITY, KOLHAPUR CENTRE FOR DISTANCE EDUCATION Diffeetial Equatios (Mathematics) Fo K M. Sc. Pat-I J

2 Copyight Pescibed fo Regista, Shivaji Uivesity, Kolhapu. (Mahaashta) Fist Editio 8 Secod Editio M. Sc. Pat-I All ights eseved, No pat of this wo may be epoduced i ay fom by mimeogaphy o ay othe meas without pemissio i witig fom the Shivaji Uivesity, Kolhapu (MS) Copies : Published by: D. D. V. Muley Regista, Shivaji Uivesity, Kolhapu-46 4 Pited by : Shi. A. S. Mae, I/c. Supeitedet, Shivaji Uivesity Pess, Kolhapu-46 4 ISBN Futhe ifomatio about the Cete fo Distace Educatio & Shivaji Uivesity may be obtaied fom the Uivesity Office at Vidyaaga, Kolhapu-46 4, Idia. This mateial has bee poduced with the developmetal gat fom DEC-IGNOU, New Delhi. (ii)

3 Cete fo Distace Educatio Shivaji Uivesity, Kolhapu Pof. (D.) N. J. Pawa Vice-Chacello, Shivaji Uivesity, Kolhapu EXPERT COMMITTEE D. D. V. Muley Regista, Shivaji Uivesity, Kolhapu. ADVISORY COMMITTEE Pof. (D.) N. J. Pawa Vice-Chacello, Shivaji Uivesity, Kolhapu. D. A. B. Rajge Diecto BCUD, Shivaji Uivesity, Kolhapu. D. B. M. Hidea Cotolle of Eamiatio Shivaji Uivesity, Kolhapu. D. (Smt.) Vasati Rasam Dea, Faculty of Social Scieces, Shivaji Uivesity, Kolhapu. Pof. (D.) B. S. Sawat Dea, Faculty of Commece, Shivaji Uivesity, Kolhapu. D. T. B. Jagtap Dea, Faculty of Sciece, Shivaji Uivesity, Kolhapu. D. K. N. Sagale Dea, Faculty of Educatio, Shivaji Uivesity, Kolhapu. D. D. V. Muley Regista, Shivaji Uivesity, Kolhapu. Shi. B. S. Patil Fiace ad Accouts Office, Shivaji Uivesity, Kolhapu. Pof. (D.) U. B. Bhoite Lal Bahadu Shasti Mag, Bhaati Vidyapeeth, Pue. Pof. (D.) A. N. Joshi Diecto, School of Educatio, Y. C. M. O. U. Nashi. Shi. J. R. Jadhav Dea, Faculty of Ats & Fie Ats, Shivaji Uivesity, Kolhapu. Pof. (D.) S. A. Bai Diecto, Distace Educatio, Kuvempu Uivesity, Kaataa. Pof. D. (Smt.) Cima Yeole (Membe Secetay) Diecto, Cete fo Distace Educatio, Shivaji Uivesity, Kolhapu. B. O. S. MEMBERS OF MATHEMATICS Chaima- Pof. S. R. Bhosale P.D.V.P. Mahavidyalaya, Tasgao, Dist. Sagli. D. L. N. Kata Head, Dept. of Mathematics, Shivaji Uivesity, Kolhapu. D. H. T. Dide Kamvee Bhauao Patil College, Uu-Islampu, Tal. Walwa, Dist. Sagli. D. T. B. Jagtap Yashwatao Chava Istitute of Sciece, Sataa. Shi. L. B. Jamale Kisha Mahavidyalaya, Rethae B., Kaad, Dist. Sataa. D. A. D. Lohade Yashavatao Chava Waaa Mahavidyalaya, Waaaaga. Pof. S. P. Pataa Viveaad College, Kolhapu. Pof. V. P. Rathod Dept. of Mathematics, Gulbaga Uivesity, Gulbaga, (Kaataa State.) Pof. S. S. Bechalli Dept. of Mathematics,Kaataa Uivesity, Dhawad. Shi. Satosh Pawa,, 'A' Wad Sadashiv Jadhav, Housig Society, Radhaagai Road, Kolhapu-46. (iii)

4 Cete fo Distace Educatio Shivaji Uivesity, Kolhapu. Diffeetial Equatios Witig Team Uit No. D. (Ms.) Saita Thaa Depatmet of Mathematics Shivaji Uivesity, Kolhapu. Mahaashta All Edito D. (Ms.) Saita Thaa Depatmet of Mathematics, Shivaji Uivesity, Kolhapu. Mahaashta. (iv)

5 Peface Lage umbes of studets appea fo M.A./M. Sc. Eamiatios eteally evey yea. I view of this, Shivaji Uivesity has itoduced the Distace Educatio Mode fo eteal studets fom the yea 7-8, ad etusted the tas to us to pepae the Self Istuctioal Mateial (SIM) fo aspiats. It is hoped that studets must lea Mathematics ot oly to become competet mathematicias but also silled uses of Mathematics i the solutio of poblems i the eal wold. They must lea how to use thei Mathematical owledge i solvig the poblems of the eal wold. Diffeetial equatios usually ae desciptio of physical systems. This boo o Diffeetial Equatios cosists of fou chaptes. Chapte oe cotais the complete discussio of liea equatios with costat coefficiets, icludig the uiqueess theoem. I chapte two liea equatios with vaiable coefficiets ae tea. Equatios with aalytic coefficiets ae itoduced ad seies solutios ae obtaied by a simple fomal pocess. A detailed teatmet of liea equatios with egula sigula poits is discussed i chapte fou. Classificatio of egula sigula poits ad egula sigula poits at ifiity is studied. I chapte five eistece ad uiqueess of solutios of fist ode iitial value poblem ae established. The iumeable eamples ad eecises ae give at the ed of each uit. The boo itoduces the studets to some of the abstact topics that pevade mode aalysis. The fist chapte deals with the Riema Stieltjes itegatio. The poblems i Physics ad Chemisty which ivolve mass distibutio that ae patly discete ad patly cotiuous ca be solved by usig Riema Stietjes itegatios. The Chapte deals with covegece ad uifom covegece of sequeces of fuctios ad seies whee as the Chapte 3 cosists of multidimesioal calculus. The Chapte 4 deals with implicit fuctios ad etemum poblems which have wide applicatios i optimizatio theoy. Lie itegals, suface itegals ad Volume itegals ae the subject matte of Chapte 5. This povides sufficiet bacgoud to study the Gauss divegece Theoem ad Stoes Theoem. (v)

6 We owe a deep sese of gatitude to the Vice-Chacello D. N. J. Pawa who has give impetus to go ahead with ambitious pojects lie the peset oe. D. Saita Thaa, Reade, Depatmet of Mathematics, Shivaji Uivesity has to be pofusely thaed fo the ovatios he has poued to pepae the SIM o Diffeetial Equatios. We also tha Pof. M. S. Chaudhay, Head, Depatmet of Mathematics, Shivaji Uivesity, Diecto of Cete fo Distace Educatio Ms. Cima Yeole ad Deputy Diecto Shi. Raj Patil fo thei help ad ee iteest i completio of the SIM. Pof. S. R. Bhosale Chaima BOS i Mathematics Shivaji Uivesity, Kolhapu-464. (vi)

7 M. Sc. (Mathematics) Diffeetial Equatios Cotets Chapte : Liea Equatios with Costat Coefficiets Chapte : Liea Equatios with Vaiable Coefficiets 53 Chapte 3 : Liea Equatios with Regula Sigula Poits Chapte 4 : Eistece ad Uiqueess of Solutio to 59 Fist Ode Equatios (vii)

8 p p p M. Sc. (Mathematics) Pape III Diffeetial Equatios D. Saita Thaa p Depatmet of Mathematics Shivaji Uivesity, Kolhapu (M.S.) Diffeetial Equatios

9 Chapte Liea Equatios with Costat Coefficiets Cotets : Uit : Iitial value poblems fo secod ode equatios. Uit : Liea depedece ad idepedecce Uit 3 : The homogeous equatio of ode Uit 4 : The o-homogeeous equatio of ode Itoductio : We live i a wold of iteelated chagig etities. The positio of the eath chages with time, the velocity of fallig body chages with distace, the bedig of a beam chages with the weight of the load placed o it, the aea of cicle chages with the size of the adius, the path of pojectile chages with the velocity ad agle at which it is fied. I the laguage of mathematics chagig etities ae called vaiables ad the ate of chage of oe vaiable with espect to aothe is called deivative. Equatios which epess a elatio amog these vaiables ad thei deivatives ae called diffeetial equatios. A Liea diffeetial equatio of ode with costat coefficiets is a equatio of the fom ( ) ( ) ( ) ay + ay + a y + + ay b( ), whee, a, a, a,, a ae comple costats ad b is comple valued fuctio o a iteval I : a< < b. The opeato L defied by ( ) ( ) ( ) L( φ)( ) φ ( ) + a φ ( ) + a φ ( ) aφ( ) is called as diffeetial opeato of ode with costat coefficiets. The equatio L(y) b() is called o-homogeous equatio. If b() fo all i I the coespodig equatio L(y) is called a homogeous equatio. Diffeetial Equatios ()

10 Uit : Iitial Value Poblems fo Secod Ode Equatios whee Hee, we ae coceed with the equatio Theoem.. Diffeetial Equatios Ly ( ) y + ay + ay a ad a ae costats. Let, a, a be costats ad coside the equatio L(y) y + a y + a y. If, ae distict oots of the chaacteistic polyomial p() + a + a the the fuctios φ ( ) e ad φ ( ) e ae solutios of L(y).. If is a epeated oot of the chaacteistic polyomial p(), the the fuctios ( ) φ e ad ( ) φ e ae solutios of L(y). Poof : Let f () e be a solutios of L(y). + + ( + a+ a) e Le ( ) ( e ) a( e ) ae L (e ) if ad oly if p() + a + a.. If ad ae distict oots of p() the L( e ) L( e ) ad φ ( ) e ad φ ae solutios of L(y).. If is a epeated oot of p() the ( ) e P( ) ( ) ad p( ) ( ) Le ( ) Pe ( ) fo all &. Le ( ) Pe ( ) [ ] Le ( ) P() + P () e. At, P( ) P ( ). i.e. Le ( ) thus, showig that e is a solutio of L(y). Thus if is a epeated oot of the chaacteistic polyomial P(), the ( ) φ e ad φ ae solutios of L(y). ( ) e Theoem.. : If f ad f ae two solutios of L(y) the C f + Cf is also a solutio of L(y). Whee, C ad C ae ay two costats. Poof : Let f ad f be two solutios of L(y) L( φ ) φ + a φ + a φ ()

11 L( φ ) φ + a φ + a φ Suppose C ad C ae ay two costats the the fuctio f defied by f C f + C f is also a solutio of L(y). L( φ) ( aφ + c φ ) + a ( aφ + c φ ) + a ( aφ + c φ ) c ( φ + aφ + a φ ) + c ( φ + aφ + a φ ) cl( φ ) + c L( φ ) The fuctio f which is zeo fo all is also a solutio called the tivial solutio of L(y). The esults of above two theoems allow us to solve all homogeeous liea secod ode diffeetial equatios with costat coefficiets. Defiitio. : A iitial value poblem L(y) is a poblem of fidig a solutio f satisfyig φ( ) α ad φ ( ) β whee, is some eal umbe ad a, b ae give costats. Theoem..3 : (Eistece Theoem) Fo ay eal ad costats a, b, thee eists a solutio f of the iitial value poblem Ly ( ) y + ay + ay, y ( ) α, y ( ) β, < <. Poof : By theoem.. thee eist two solutios f ad f that satisfy L(f ) L(f ). Fom theoem.. we ow that c f + c f is a solutio of L(y). We show that thee ae uique costats c, c such that φ c φ + cφsatisfies φ( ) α ad φ ( ) β. ad φ( ) cφ ( ) + c φ ( ) α φ ( ) cφ ( ) + c φ ( ) β Above system of equatios will have a uique solutio c, c if the detemiat φ( ) φ( ) φ( ) φ ( ) φ( ) φ ( ). φ ( ) φ ( ) By theoem.. (), φ ( e ) ad φ ( e ), ae two solutio of Ly ( ) fo ( ) e + e e e e ( ). By theoem.. (), φ ( ) e ad φ ( ) e, ae solutios of L(y) ad Diffeetial Equatios e e + e e e e (3)

12 Thus, the detemiat coditio is satisfied i both the cases. Theefoe, c, c ae uiquely detemied. The fuctio f c f + c f is a desied solutio of the iitial value poblems. Defiatio. : A solutio of a diffeetial equatio will be called a paticula solutio if it satisfies the equatio ad does ot cotai abitay costats. Theoem..4 : Let, f be ay solutio of Whee, Ly ( ) y + ay + ay o a iteval I cotaiig a poit, The fo all i I. φ( ) e φ( ) φ( ) e φ( ) φ( ) + φ ( ) ad + +. Poof : Let, u ( ) φ( ) φ( ) + φ ( ) φ( ) φ( ) + φ ( ) φ ( ) The, u ( ) φ ( ) φ( ) + φ( ) φ ( ) + φ ( ) φ ( ) + φ ( ) φ ( ) ad u ( ) φ( ) φ ( ) + φ ( ) φ ( ) Sice f is a solutio of L(y), L( φ) φ + aφ + aφ as φ( ) φ( ) i.e. φ ( ) aφ ( ) aφ( ) ad the above iequality becomes u ( ) φ( ) φ ( ) + φ ( ) a φ ( ) + a φ( ) [ ] [ ] φ φ φ + a ( ) ( ) + a ( ) But, φ( ) φ ( ) φ( ) + φ ( ) Theefoe, Thus, we get ( ) φ ( ) ( a a ) φ φ u ( ) + a + a ( ) + + a φ( ) + + ( ) + ( ) u( ) u ( ) u ( ) u ( ) u ( ) u( ) is equivalet to u ( ) u( ) sice epoetial fuctios ae positive o multiplyig above iequality by e we get Diffeetial Equatios (4)

13 ( ) ( ) e u u e u ( ) ( ) ( ). Itegatig above iequality betwee the limits to fo > yields. ( ) ( ) e u e u ( ) u ( ) e u ( ) ( ) Thus, φ( ) e φ( ) Similaly, fo > the iequality u() u () implies ( ) ( ) ( ) φ e φ Theefoe fo > we get ( ) ( ) φ( ) e φ( ) e φ( )... (..) Fo <, the sig of above iequality will get chaged ( ) ( ) φ( ) e φ( ) e φ( ) This iequality ca be witte as ( ) ( ) φ( ) φ( ) φ( ) e e sice <, >. ( ) ( ) φ( ) φ( ) φ( ) e e... (..) Equatio (..) ad (..) togethe ca be put i the fom φ( ) φ( ) φ( ) e e Sice all the tems i above iequality ae positive the squae oot of each tem esults ito the equied iequality. Theoem..5 (Uiqueess Theoem) Let a, b be ay two costats ad let be ay eal umbe. O ay iteval I cotaiig thee eists at most oe solutio f of the iitial value poblem Ly ( ) y + ay + ay, y ( ) α, y ( ) β Poof : Suppose f ad y ae two solutios. Let θ φ ψ. Sice L( φ) L( ψ), L( θ) L( θ ψ) L( θ) L( ψ) Sice φ( ) ψ( ) α ad φ ( ) ψ ( ) β, θ( ) φ( ) ψ( ) ad θ ( ) φ( ) ψ ( ) Thus, L( θ), θ( ) ad θ ( ). θ( ) θ( ) + θ ( ) Diffeetial Equatios (5)

14 By theoem (..4) we see that θ( ) θ( ) + θ ( ) fo all i I This implies q () fo all i I. But θ( ) θ( ) ψ( ) i.e. φ( ) φ( ). Theoem..6 : Let f, f be two solutios of L(y) give by theoem... If c, c ae ay two costats the fuctio f c f + c f is a solutio of L(y) o < <. Covesely, if f is ay solutio of L(y) o < <, the thee ae uique costats C ad C such that f C f + C f. Poof : Fist pat of the theoem follows fom theoem... Covesely suppose f is ay solutio of L(y). Let φ( ) α ad φ ( ) β fo some costats a ad b. I the poof of eistece theoem..3 we showed that thee is a solutio y of L(y) satisfyig. ψ( ) α, ψ ( ) β of the fom ψ( ) cφ( ) + cφ( ) whee c ad c ae uiquely detemied by a ad b. By uiqueess theoem..5 φ ψ, fo all. Eamples :. Fid all solutios of the followig equatios. (a) y² 4 y (b) y²+ i y + y (c) y² 4 y + 5y Aswe : (a) The chaacteistic polyomial is p() 4. ad ae two distict oots of p (). Theefoe φ() s e ad φ() e ae two solutios. Fo ay costats c ad c, c e + c e is a solutio. Thus the geeal solutio is φ ( ) ce + ce. (b) The chaacteistic polyomial p() + i + p () i± () i 4 ± 8 i i± i ( ) ± i Diffeetial Equatios (6)

15 Thus ( + ) ad ( ) i i ae two distict oots of p(). ( + ) i ( ) Theefoe φ( ) ad φ( ) costats c ad c, ( + ) i ( ) i e e ae two solutios. Thus, fo ay φ ( ) c e + c e is a geeal solutio. (c) The chaacteistic polyomial p() p() gives + i ad i as two distict oots. f () e ( + i) ad f () e ( i) ae two solutios of the diffeetial equatio. Fo ay costats c ad c, f () c e ( i) + c e ( + i) is a geeal solutio. I paticula fo c c we get, i i e + e φ ( ) e e cos. ad fo c ad c i i we get i i e e φ ( ) e e si i Thus, f () A e cos + B e si is a solutio of the diffeetial equatio fo ay costats A & B.. Fid the solutios f of the followig iitial value poblems. (a) φ + φ 6φ, φ(), φ () (b), (), π φ + φ φ φ (c) φ + φ, is ay costat, φ(), φ ( π) (d) φ φ 3φ, φ(), φ () Aswe : (a) The chaacteistic polyomial p() + 6. ad 3 ae distict oots 3 φ ( ) c e + c e is a geeal solutio. φ () c + c... () 3 φ () φ ( ) c e 3c e at, gives φ () c 3c... () solvig equatio () ad () fo c ad c we get c 3/5 ad c +/5. 3 3e e Thus, the equied solutio is φ ( ) i Diffeetial Equatios (7)

16 (b) The chaacteistic polyomial is p () +. i ad i ae distict oots φ ( ) c cos+ c siis a geeal solutio. φ () c cos + c si gives c π c π + c π φ ( ) cos si. gives c. Thus, f () cos + si is the equied solutio. (c) The chaacteistic polyomial is p () + sice is ay costats, ca be positive, egative o zeo. Case. > The i ad i; ae distict oots. i i φ ( ) c e + c e is a geeal solutio I geeal φ ( ) Acos + Bsi is a solutio. φ () Acos + Bsi i.e. A φπ ( ) Acos π+ Bsiπ i.e. A Thus, f () B si is a solutio whee B is ay costat. Case. p () a epeated oot. φ( ) c e + c e c + c is a solutio φ() c φπ ( ) c+ cπ c Theefoe thee is o otivial solutio coespodig to. Case 3. < fo, p () + has distict oots & ( Sice <, > ) c φ ()c e + e φ () c + c π ce π φπ ( ) ce + Simultaeous evaluatio of above two equatios give c c. Thus, thee is o o-tival solutio coespodig to <. The oly o-tivial solutio fo the give equatio is φ( ) Bsi. (d) The chaacteistic polyomial p() 3 3, ae two distict oots. Diffeetial Equatios (8)

17 \ 3 ( ) ce ce φ + is a geeal solutio φ() φ() c + c 3 φ ( ) 3 c e c e φ () φ () 3 c c Thus, c + c ad 3c c gives c ad c Theefoe φ ( ) e e is the equied solutio Fill i the blas. EXERCISES (i) If, ae distict oots of chaacteistic polyomial p () + a + athe φ ( )... ad φ ( )... ae solutios of the diffeetial equatio y + a y + a y (ii) If p () ( ) is a chaacteistic polyomial the φ ( )... ad φ ( )... ae two solutios of the diffeetial equatio y y + y. (iii) Uiqueess theoem states that... (iv) Solutio of y y + 4y ae φ( )... ad φ( ).... (v) The geeal solutio of y 3y + y is.... Fid the getal solutio of each of the followig equatio. (i) y + 4y (ii) y y (iii) y + y 6y (iv) y + 4 y y (v) y ay + a y (vi) y 4y + y 3. Fid the solutio of the followig iitial value poblems : (i) y, y(), y () (ii) y + 4y + 4y, y(), y () (iii) y y + 5y, y(), y () 4 (iv) y y y y( π ) y ( π ) Aswes : 4 +,,. (i) φ ( ) e, φ ( ) e (ii) φ ( ) e, φ ( ) e Diffeetial Equatios (iii) theoem..5 (iv) φ ( ) e, φ ( ) e (v) ce + ce (9)

18 . (i) 4 c + c e (ii) ce + c e (iii) 3 ce + c e (iv) 6 ce + c e a (v) ( c+ c) e (vi) e ( ccos4+ csi4 ) 3. (i) 3 (ii) ( + 3) e (iii) e (cos+ si ) (iv) e 4 π si 4 Uit : Liea Depedece ad Idepedece Evey solutio of the equatio L (y) is a liea combiatio of two solutios obtaied i theoem... Theefoe these two solutios spa the solutio space of the diffeetial equatio L(y). Defiatio.3 : A set of eal o comple fuctios f, f, f 3,..., f defied o a iteval (a, b) is said to be liealy idepedet whe c f ( ) + c f ( ) + c 3 f 3 ( ) + + c f( ) fo evey i (a, b) implies c c c3 c. Defiatio.4 : Give the fuctios f, f, f 3,, f if costats c, c, c 3,, c ot all zeo eist such cf ( ) + c f ( ) + cf 3 3 ( ) + + cf( ) fo evey i (a, b), the these fuctios ae liealy depedet. A set which is ot liealy idepedet is said to be liealy depedet. Thee ae two otios of liea idepedece, accodig as we allow the coefficiets c,,, 3,..., to assume oly eal values o also comple values. I the fist case, oe says that the fuctios ae liealy idepedet ove the field of eals; i the secod case, that they ae liealy idepedet ove the comple field. Lemma.. : A set of eal valued fuctios o a iteval (a, b) is liealy idepedet ove the comple field if ad oly if it is liealy idepedet ove the eal field. Poof : If the set of eal valued fuctios o a iteval (a, b) is liealy idepedet ove the comple field the it is liealy idepedet ove the field of eals. Covesely suppose the set is liealy idepedet ove the eal field. Theefoe fo j j j 3 3 j cj f j( ) j α R, α f ( ) α f ( ) + α f ( ) + α f ( ) + + α f ( ) fo all i (a, b) implies a j fo all j,, 3...,. Let fo all i (a, b) ad fo some c C, j,,3,. Sice the fuctio f j ae eal valued ad c f ( ), j * * c * j cj ( ) cj fj. implies cj fj( ). Thus, * f j( ). But j j ( cj cj )/ i i ae all eal ad the set is liealy idepedet ove the eal field theefoe c c *. But the c j s j j j j Diffeetial Equatios ()

19 ae all eal theefoe c f ( ) implies c j fo j,,... j j j A set of fuctios which is liealy depedet o a give domai may become liealy idepedet whe the fuctios ae eteded to a lage domai. Howeve, a liealy idepedet set of fuctios clealy emai liealy idepedet o the esticted domai. Illustatio : The fuctios f ad f defie by f () Cos ad f () Si ae liealy idepedet o the eal lie IR ad theefoe ae liealy idepedet o (, p). Illustatio : The fuctios f ad f defie by f (), f () ae liealy idepet o the iteval (, ) but is ot liealy idepedet o the iteval (, ) as o the iteval (, ), f () f (). Theoem.. : Let a, a be costats ad coside the equatio Ly ( ) y + ay + ay. The two solutios of L (y) give i the theoem.. ae liealy idepedet o ay iteval I. Case. Poof : Let, be the oots of chaacteistic polyomial p() + a + a. If ¹, the ( ) φ e ad ( ) φ e ae two solutios of the equatio L(y) o a iteval I. I. c, ( ) ce Suppose ce + c e fo all i I. The c + fo all i I. ( ) Diffeetiatio of above equatio with espect to givesc ( ) e fo all i Sice, ¹ ad epoetial fuctio i o-zeo, c is zeo. But if c is zeo the ( ) ce + implies c is zeo. Thus, ce + c e implies c c. Case. Theefoe φ ( ) e ad φ ( ) e ae liealy idepedet. If, the ( ) φ e ad ( ) φ e ae two solutios of the equatio L(y) o a iteval I. e Suppose ce + ce the c+ c fo all i I. Theefoe c c. Thus, f ad f ae liealy idepedet Thus, i both cases the two solutios f ad f of L(y) ae liealy idepedet. Defiatio.5 : Assume that each of the fuctios f ( ), f ( ), f 3 ( ),, f( ) ae diffeetiable atleast ( ) times i the iteval (a, b). The the detemiat Diffeetial Equatios ()

20 Diffeetial Equatios f( ) f( ) f3( ) L f( ) f ( ) f ( ) f 3 ( ) L f ( ) f ( ) f ( ) f 3 ( ) L f ( ) M M M M ( ) ( ) ( ) ( ) 3 L f ( ) f ( ) f ( ) f ( ) deoted by W( f, f, f3,..., f)( ) is called the wosia of the fuctios f, f, f3,..., f. Theoem.. : Two solutios f, f of L (y) ae liealy idepedet o a iteval I if ad oly if W( φ φ ) ( ) fo all i I., Poof : Suppose W( φ, φ )() fo all i I Let c, c be costats such that c f () + c f () fo all i I. The c f () + c f () fo all i I. Above two equatios ca be witte as φ( ) φ( ) c φ ( ) φ ( ) c Sice, W( φ, φ)( ) fo all i I, the coefficiet mati is ivetible. O pemultiplyig the ivese of the coefficiet mati esults i c c. This poves that f ad f ae liealy idepedet o I. Covesely, assume that f, f ae liealy idepedet o I. Suppose that thee is a poit i I such that W( φ, φ ) ( ). The the system of equatios φ( ) φ( ) c φ ( ) φ ( ) c has a solutio c, c whee at least oe of these umbes is ot zeo. Let c, c, be such a solutio ad coside the fuctio ψ( ) cφ( ) + cφ( ). Now L( ψ) ad ψ( ), ψ ( ). Theefoe ψ( ) ψ( ) + ψ ( ). By theoem..4 ψ ( ). But ψ( ) ψ( ) + ψ ( ). Theefoe ψ ( ) fo all i I ad thus c φ ( ) + c φ ( ) fo all i I. But the f ad f ae liealy depedet. Thus, the suppositio W( φ, φ ) ( ) must be false ad theefoe W( φ, φ) ( ) fo all i I. I the et theoem we will pove that we eed to compute W( φ, φ) at oly oe poit to test the liea idepedece of the solutios f ad f. ()

21 Theoem..3 : Let f, f be two solutio of L(y) o a iteval I ad let be ay poit i I. The two solutios f ad f ae liealy idepedet o I if ad oly if W( φ, φ) ( ). Poof : If f ad f ae liealy idepedet o I the by theoem.., W( φ, φ) ( ) fo all i I. I paticula W( φ, φ) ( ) covesely, suppose W( φ, φ) ( ) ad suppose c, c ae costats such that c φ ( ) + c φ ( ) fo all i I. The i.e. c φ ( ) + c φ ( ) ad cφ ( ) + cφ ( ). φ( ) φ( ) c φ ( ) φ ( ) c But sice the detemiat of the coefficiet is W( φ, φ) ( ) we obtai c c. Thus f, f ae liealy idepedet o I. I the et theoem we show that the owledge of two liealy idepedet solutios of L(y) is sufficiet to geeate all solutios of L(y). Theoem..4 : Let f, f be ay two liealy idepedet solutios of L(y) o a iteval I. Evey solutio f of L(y) ca be witte uiquely as φ c φ + c φ whee c, c ae costats. Poof : Let be a poit i I. Let φ( ) α, φ ( ) β. Sice f, f ae liealy idepedet o I we ow that W( φ, φ)( ). Coside the two equatios. φ( ) φ( ) c α φ ( ) φ ( ) c β Sice W( φ, φ) ( ), above system of equatios has a uique solutio c, c. Fo this choice of c, c the fuctioψ( ) cφ( ) + cφ( ) satisfies ψ( ) cφ( ) + cφ( ) α φ( ) i.e. ψ( ) φ( ) similaly ψ ( ) φ ( ) ad L( ψ). Fom the uiqueess theoem..5 it follows that ψ φ o I i.e. φ c φ + c φ. Eamples : Q. Show that the fuctios e, e, e 3 ae liealy idepedet. As. : Method : Let 3 ce + c e + ce 3 the c+ ce + c3e... () Diffeetial Equatios (3)

22 Diffeetiate above equatio () with espect to the c e + c 3 e implies c + c e... () 3 By diffeetiatig equatio () with espect to we get ce 3 theefoe c 3. But the by equatio () c ad by equatio () we get c. Thus c c c 3. Theefoe the fuctios e, e, e 3 ae liealy idepedet. Method : 3 Let 3 φ ( ) e, φ ( ) e, φ ( ) e 3 e e e 3 3 W( φ, φ, φ 3) e e 3e e e e 3 3 e 4e 9e 4 9 e 6 [(8 ) (9 3) + (4 )] e 6 ¹. by theoem.. f, f, f 3 ae liealy idepedet. Q. : The fuctios f, f ae defied o < <. Detemie whethe they ae liealy depedet o idepedet thee. (i) φ( ), φ( ) e, is a comple costat (ii) φ ( ), φ ( ) 5 (iii) φ( ), φ( ) (iv) φ( ) cos, φ( ) si As. (i) : Method : c ( ) + c ( ) Let φ φ i.e. c + c e... () if, c+ c fo all Rimplies c ad c. \ f, f ae liealy idepedet if ¹, diffeetiate equatio () with espect to the c+ ce Agai diffeetiate above equatio with espect to the ce. But ad e theefoe c ad fom equatio () we get c. Thus f, f ae liealy idepedet. Method : e W( φ, φ ) e e Diffeetial Equatios (4)

23 e ( ) fo IR \ f, f ae liealy idepedet Method 3 : idepedet. As. (ii) : W(, ) () Let c φ + c φ i.e. c + c 5 if ( c + 5 c ) φ φ theefoe by theoem..3 f, f ae liealy If we choose c 5c ¹ the the liea combiatio c φ + c φ theefoe by defiitio.4, f, f ae liealy depedet. As. (iii) : > c φ + c φ ( c + c ) as Fo ad fo < cφ+ cφ ( c c) as Thus, cφ+ cφ fo R ( c + c ) ad ( c c ) fo evey R above two equatios hold tue if ad oly if c c. Thus f, f defied by φ ( ) ad φ ( ) ae liealy idepedet. As. (iv) : φ ( ) cos ; φ ( ) si cos si W( φ, φ ) ( ) si cos Q W( φ, φ )( ), φ, φ ae liealy idepedet. Q3. : Let f be ay fuctio satisfyig the bouday value poblem y + y, y() y( π), y () y ( π),,,,3,... show that π φ ( ) φ ( ) d if m. m As. : The chaacteistic polyomial p () + has oots i, i ad theefoe the Diffeetial Equatios (5)

24 geeal solutio φ ( ) ccos + dsi Fom the give bouday coditios. φ () c ad φ ( π) c φ () φ ( π) ad φ () d ad φ ( π) d φ () φ ( π) Thus, φ ( ) c cos + d si satisfies the give bouday coditios. The solutio f satisfies φ ( ) + φ ( ) whee as φ ( ) + m φ ( ) holds. Thus, ( m ) φ ( ) φ ( ) φ ( ) φ ( ) φ ( ) φ ( ) m m m Itegatig above equatio fom to p We get, ( ) π φ ( ) φm( ) φ( ) φ m ( ) π φ φm φ φ φ φ m m m ( ) ( ) d ( ) ( ) ( ) ( ) d φ ( ) φm( ) φ( ) φ m ( ) But φ () c, φ ( π) c ; φ () d, φ ( π) d Similaly, φ (), ( ) ; () m cm φm π cm φm mdm φm ( π) π Thus, ( m ) φ ( ) φ ( ) d [ d c c md ] [ d c c md ] Sice, π φ m m π m m m m m m, ( ) φ ( ) d. m Q4. (a) : Show that f () Si satisfies the bouday value poblem y² + y, y ( ), y ( p ),,... (b) : Usig (a) show that π si si m d if m As. 4(a) : Method : The chaacteistic polyomial p () + has oots ± i ad theefoe the geeal solutio Diffeetial Equatios (6)

25 φ ( ) c cos + d si y() φ () φ () c y( π) φ( π) φ( π) c( ). Thus, φ ( ) si is a solutio fo,, 3,... Method : φ ( ) si, φ ( ) cos φ ( ) si Thus, φ ( ) + φ ( ) si+ si. Sice, φ ( ) si satisfies φ ( ) + φ( ) ad φ (), φ ( π) φ ( ) si is a solutio of As. 4(b) : Woig o the simila lie as i eample we get, π π φ φ m y + y, y() y( π). ( m ) ( ) ( ) d ( m ) si sim d [ si ( mcos m) si m( cos ) ] π (as si si p ) Sice π φ m, ( ) φ ( ) d. m Q5 : Suppose f, f ae liealy idepedet solutios of the costat coefficiet equatio y + a y + a y Let W (f, f ) be abbeviated to W. Show that W is costat if ad, oly if a. As. : φ W φ W( φ, φ) ( φφ φφ ) φ φ The W ( ) φφ φ φ φφ + φ φ φ φ φ φ φφ φφ But f ad f ae solutios of y + ay + ay. Diffeetial Equatios (7)

26 + a + a a a Theefoe φ φ φ φ φ φ Similaly, φ aφ aφ W ( ) ( ) Thus, φ aφ aφ φ aφ aφ a ( φφ φ φ ) aw Thus, W iff a Theefoe W costat if ad oly if a Q6 : Let f, f be two diffeet fuctio o a iteval I, which ae ot ecessaily solutios of a equatio L(y). Pove the followig As. 6(a) : (a) If f, f ae liealy depedet o I the W(f, f ) () fo all i I (b) If W(f, f ) ( ) ¹ fo some i I, the f, f ae liealy idepedet o I. (c) W(f, f )() fo all i I does ot imply that f, f ae liealy depedet o I. (d) W(f, f ) () fo all i I ad f () ¹ o I, imply that ae f, f liealy depedet. Suppose f, f ae liealy depedet o I the c φ ( ) + c φ ( ) fo some o-zeo c ad c. i.e. As. 6(b) : c φ ( ) ( ). φ c φ φ W( φ, φ )( ) φ ( ) φ ( ) φ ( ) φ ( ) φ φ W( φ, φ c c )( ) ( ) ( ) ( ) ( ) c φ φ φ c φ W( φ, φ )( ) fo all I. Suppose cφ( ) + cφ( ) the c φ ( ) + c φ ( ) Thus we have a system of equatio φ( ) φ( ) c φ ( ) φ ( ) c Diffeetial Equatios (8)

27 Theefoe at φ( ) φ( ) c φ ( ) φ ( ) c Thus, c c if ad oly if the coefficiet mati is ivetible i.e. the detemiat of coefficiet mati is o-zeo As. 6(c) : φ( ) φ( ) But W( φ, φ)( ) φ ( ) φ ( ) Sice, W( φ, φ)( ) c c cφ( ) + cφ( ) c c. Hece f ad f ae liealy idepedet o I. Defie φ ( ), φ ( ) fo >, φ ( ), φ ( ) W( φ, φ)., ( ) ( ) W(, ) fo φ φ φ φ fo <, φ ( ) ad φ ( ) W( φ, φ). Thus W( φ, φ ) ( ) fo < < c ( ) + c ( ) Let φ φ fo >, c φ ( ) + c φ ( ) ( c + c ). c +... (i) c fo <, c φ ( ) + c φ ( ) c c. c c... (ii) But c c c c c c + ad Thus, cφ+ cφ c c Theefoe f, f ae liealy idepedet. Diffeetial Equatios (9)

28 As. 6(d) : φ( ) φ( ) W( φ, φ)( ) W( φ, φ)( ) φ ( ) φ ( ) φ ( ) φ ( ) φ ( ) φ ( ) φ ( ) φ ( ) φ ( ) φ ( ) Sice φ ( ) I φ ( ) φ ( ) φ ( ) φ ( ) φ ( ) φ φ costat (say) φ φ Theefoe φ ( ) φ ( ) ad hece φ, φ ae liealy depedet. Q7 : If f, f ae two solutio of L(y) o a iteval I cotaiig a poit, the a( ) e W( φ, φ )( ) W( φ, φ )( ). As. : Sice f, f ae solutio of L(y), φ φ φ + a + a φ φ φ + a + a O multiplyig the fist equatio by f, secod equatio by f ad addig we obtai Let φφ φ φ + a ( φφ φφ ) + a ( φφ φφ) ( φφ φ φ ) + a ( φφ φφ )... (i) φ( ) φ( ) W W ( φ, φ) ( ) φ ( ) φ ( ) The φ φ φ φ W ( ) ( ) ( ) ( ) W ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ad φ φ + φ φ φ φ φ φ φ( ) φ ( ) φ( ) φ ( ) Thus, equatio (i) becomes W + aw. Thus W satisfies the fist ode diffeetial equatio W + aw Diffeetial Equatios ()

29 a Hece, W( ) c e whee c is costat of itegatio. At we get a, a W( ) c e i.e. c e W( ) a a Thus, W( ) e W( ) e a ( ) W( ) e a ( ), Theefoe W( φ φ )( ) e W( φ, φ )( ) EXERCISES. The fuctios f, f ae defied o < < Detemie whethe they ae liealy depedet o idepedet thee. (i) φ( ) cos, φ( ) si (ii) φ( ) si, φ( ) e i (iii) φ ( ) si, φ ( ) cos (iv) φ( ), φ( ) cos (v) φ ( ) si, φ ( ) cos (vi) φ ( ), φ ( ) si, φ ( ) cos 3 (vii) i φ ( ) cos, φ ( ) e + e i. State whethe the followig statemets ae tue o false. (a) If f, f ae liealy idepedet fuctios o a iteval I, they ae liealy idepedet o ay iteval J cotaied iside I. (b) If f, f ae liealy depedet o a iteal I, they ae liealy depedet o ay iteal J cotaied iside I. (c) If f, f ae liealy idepedet solutios of L (y) o a iteal I, they ae liealy idepedet a ay iteal J cotaied iside I. (d) If f, f ae liealy depedet solutios of L (y) o a iteval I, they ae liealy depedet o ay iteal J cotaied iside I. As. :. (i) idepedet (ii) idepedet (iii) idepedet (iv) idepedet (v) idepedet (vi) depedet (vii) depedet. As. :. (a) false (b) tue (c) tue (d) tue S Diffeetial Equatios ()

30 Uit 3 : The Homogeeous Equatio of Ode Eveythig we have doe fo the secod ode equatio ca be caied ove to the case of the equatio of ode. Hee, we ae coceed with the equatio ( ) ( ) ( ) Ly ( ) y + ay + ay + + ay, whee, a, a, a3,..., a ae costats. Theoem.3. : Let,, 3,..., s be the distict oots of the chaacteistic polyomial p () + a + a + + a ad suppose i has multiplicity mi( m+ m+ m3+ + ms ). The fuctios m m e, e,..., e ; e, e,..., e ;...;, m e, e, e,..., e s s s s s ae solutios of ( ) ( ) ( ) Ly ( ) y + ay + ay + + ay Poof : Suppose i is a oot of p() of multiplicity m i. The () ( ) i p i qwhee () q is a polyomial of degee m i. O diffeetiatig p(), (m i ) times we get, ad so o m i i i m () ( ) i () + ( ) i p q m q() [ ] m i i i i ( ) q( )( ) + m q( ) mi mi mi i i i i i i m i i i i i i i p ( ) ( ) q ( ) + m ( ) q ( ) + m ( m )( ) q( ) ( ) ( ) q () + m ( ) q() + m ( m ) q() m ( ) i Polyomial of ode m i [ ] mi ( m ) i [ i] [ ] ( mi p ) ( ) ( ) Polyomial of ode m i ( ) Polyomial of ode m i Theefoe, ( m ) p ( ) p ( ) p ( ) p i ( ). i i i i Let e be a solutio of L(y). We see that Le ( ) pe ( ) whee p () + a. + a + + a Theefoe ( i ) ( ) i Le p e. Thus i e is a solutio of L(y). i i i m If we diffeetiate Le ( ) pe ( ) times with espect to we obtai Diffeetial Equatios ( ) Le ( ) L e L e ()

31 ( ) ( ) ( ) ( ) p () + p () + p () + + p()! e Thus fo i ad,,,...m i we get ( i Le ). Theefoe i e,,,,... m, ae solutios of L(y). This is tue fo evey chaacteistic oot i Diffeetial Equatios i with multiplicity m i. i.e. i e,,,,... mi, i,,3,... sae solutios of L(y) ad the esult follows. Theoem.3. : The solutios of L(y) give i theoem.3. ae liealy idepedet o ay iteval I. Poof : We pove that fuctios give i theoem.3. satisfy the coditio give i defiatio.3. Suppose we have costats c, i,... s, j,... m Such that Defie ij m ( ce + c e + c e c( m ) e ) m +( ce + c e + c e c( m ) e ) s s s ms s ( cs e c s e c s e c s( ms ) e ) i( ) i i i... i( mi ) i m p c + c + c + + c i The ( ) + ( ) + 3( ) ( ) s p e p e p e ps e. Assume that ot all costats c ij ae zeo. The thee will be at least oe of the polyomials p i which is ot idetically zeo o I. Suppose p s () is ot idetically zeo o I. O dividig above equatio by e we get ( ) 3 ( 3 ) ( s ) s p ( ) + p ( ) e + p ( ) e p ( ) e. Upo diffeetiatig above equatio sufficietly may (at most m i ) times, we obtai the epessio of the fom ( ) ( 3 ) ( s ) 3 s ( ) ( ) ( ) + 3( ) ( ) s s Q ( ) e + Q ( ) e Q ( ) e i.e. 3 Q Q e Q e whee the Q i s ae polyomials, degee of Q i is equal to degee of P i ad Q s does ot vaish idetically. Cotiuig this pocess we fially aive at a situatio whee, ( ) s Rs e, o I ad R s is a polyomial, degee of R s is equal to degee of P s, which does ot vaish idetically o I. But ( ) s Rs e implies Rs( ) is a cotadictio. Theefoe ou suppositio that Ps( ) is ot idetically zeo is ot tue. Thus Ps( ) fo all i I. (3)

32 Thus all costatsc ij povig that the solutios give i theoem 3. ae liealy idepedet o a iteval I. * Iitial value poblem fo th ode equatios. The poblem of fidig a solutio f of Diffeetial Equatios ( ) ( ) ( ) Ly ( ) y + ay + ay ay satisfyig φ( ) α, φ ( ) α,..., φ ( ) α whee a, a, a 3,..., a ad α, α, α3,..., α ae costats is deoted by ( ) Ly ( ), y ( ) α, y( ) α,..., y ( ) α ad is called a iitial value poblem. Theoem.3.3 : Let f be ay solutio of ( ) ( ) ( ) Ly ( ) y + ay + ay ay o a iteval I cotaiig a poit. The fo all i I ( ) φ( ) e φ( ) φ( ) e whee, + a + a + a a ad ( ) φ ( ) φ ( ) + φ ( ) φ ( ) Poof : This poof is simila to the poof of theoem..4. Let u ( ) φ( ) ( ) φ + φ φ ( ) ( ) φφ + φ φ φ φ Hece ( ) ( ) ( ) ( ) u ( ) φφ + φφ + φ φ + φφ φ φ + φ φ Theefoe u ( ) φ( ) φ ( ) + φ φ φ φ Sice f is solutio of L(y), L(f ) ad theefoe ( ) ( ) ( ) ( 3) 3 φ a φ a φ a φ... a φ O substitutig the epessio fo ( ) φ we get ( ) ( ) ( ) ( ) u ( ) φ φ + φ φ φ φ ( ) ( ) ( ) ( ) + a φ + a φ φ a φ φ ( a b ) a + b a b ( ) ( ) u ( ) ( φ + φ ) + ( φ + φ ) ( φ + φ ) ( ) ( ) ( ) a φ φ a φ φ + ( + ) ( + ) (4)

33 φ φ φ ( + a ) + (+ + a ) + ( + a ) ( ) ( ) φ φ ( + a ) + ( + a + a a ) Sice each coefficiet o the ight had side is less tha we have ( ) u ( ) ( φ + φ φ ) φ( ) u( ) Theefoe u ( ) u( ) Thus, we get u( ) u ( ) u( ) u u( ) implies ( e u( )) Itegatig above iequality betwee the limits to fo > yields ( ) ( ) e u e u ( ) i.e. ( ) u e u ( ) ( ) Thus, φ( ) e φ( ) Similaly fo > the iequality u( ) u ( ) implies ( ) ( ) ( ) φ e φ Combiig the above two iequalities we get the equied esult fo >. Fo < itechage the ole of ad ( ) ( ) We get φ( ) ( ) ( ) e φ φ e φ( ) ( ) ( ) ad φ( ) e φ( ) φ( ) e φ( ) ( ) ( ) Thus, φ( ) e φ( ) e φ( ), ( < ) which is the equied esult fo < Theoem.3.4 (Uiqueess theoem) Let α, α, α3,..., α be ay costats ad let be ay eal umbe. O ay iteval I cotaiig thee eists at most oe solutio f of L (y) satisfyig φ( ) α, φ ( ) α, ( ),..., φ ( ) α Poof : Suppose f ad y wee two solutios of L (y) o I satisfyig the above coditios at. i.e. Defie θ φ ψ Thus ( ) ( ) φ( ) ψ( ) α, φ ( ) ψ ( ) α,..., φ ( ) ψ ( ) α. Sice f ad y satisfy L( φ) L( ψ) theefoe L ( θ ) ad ( ) ( ) θ( ) φ( ) ψ( ), θ ( ),..., θ ( ). θ( ) θ ( ) + θ ( ) θ ( ) Diffeetial Equatios (5)

34 Applyig theoem.3.3 we obtai θ ( ) fo all i I. This implies θ ( ) fo all i I. i.e. φ( ) ψ( ) fo all i I. Theoem.3.5 φ, φ, φ,... φ, ae solutios of L(y) o a iteval I, they ae liealy idepedet If 3 if ad oly if W( φ, φ, φ3,... φ)( ) fo all i I. (defiitio.5) Poof : The poof is etiely simila to the poof of theoem.. Suppose W( φ, φ, φ3,... φ)( ) fo all i I. Let c, c, c 3,..., c be costats such c φ ( ) + c φ ( ) c φ ( ) fo all i I. that By diffeetiatig above equatio ( ) times we get a system of equatios as follows. φ ( ) φ ( ) φ ( ) L φ ( ) 3 c φ ( ) φ ( ) φ 3 ( ) L φ ( ) c φ ( ) φ ( ) φ3 ( ) L φ ( ) 3 c M M M M M M ( ) ( ) ( ) ( ) φ ( ) φ ( ) φ3 ( ) φ ( ) c The coefficiet mati is ivetible because the detemiat of coefficiet mati is (defiitio.5) W( φ, φ, φ3,... φ)( ). O pemultiplyig the ivese of the coefficiet mati we get, c c c3... c. This poves that φ, φ, φ3,... φ ae liealy idepedet. Covesely, assume that φ, φ,... φ ae liealy idepedet o I. Suppose thee is a poit i I such that W( φ, φ, φ3,... φ )( ). The the system of equatios φ ( ) φ ( ) φ ( ) L φ ( ) 3 c φ ( ) φ ( ) φ 3 ( ) L φ ( ) c φ ( ) φ ( ) φ3 ( ) L φ ( ) 3 c M M M M M M ( ) ( ) ( ) ( ) φ ( ) φ ( ) φ3 ( ) φ ( ) c has a solutio c, c, c3,..., c whee at least oe of these umbes is ot zeo. Let c, c,..., c be such a solutio ad coside a fuctio ψ( ) c φ ( ) + c φ ( ) c φ ( ). Now Lψ ( ) ad ( ) ψ ( ) ψ ( )... ψ ( ). Theefoe ψ ( ). But the by theoem.3.3, ψ ( ), fo all i I. Theefoe Diffeetial Equatios (6)

35 by defiatio of ψ ( ), ψ ( ) fo all i I. But the φ, φ, φ3,... φ ae liealy depedet. Thus the suppositio W( φ, φ, φ3,... φ )( ) must be false. Theefoe W( φ, φ, φ,... φ)( ) fo all i I. 3 Theoem.3.6 (Eistece Theoem) Let α, α, α3,..., α be ay costats ad let be ay eal umbe. Thee eists a solutio f of L(y) o < < satisfyig ( ) 3 φ( ) α, φ ( ) α, φ ( ) α,..., φ ( ) α Poof : Let φ, φ, φ 3,... φ be ay set of liealy idepedet solutios of L(y) o < <. We will show that thee eist uique costats c, c, c3,..., c such that φ c φ + c φ + c φ + + c φ is a solutio of L(y) satisfyig the give iitial coditios () i φ ( ) αi, i,,,...,. These costats c, c, c3,..., c would have to satisfy φ ( ) φ ( ) φ ( ) L φ ( ) 3 c α φ ( ) φ ( ) φ 3 ( ) L φ ( ) c α φ ( ) φ ( ) φ3 ( ) L φ ( ) 3 c α3 M M M M M M ( ) ( ) ( ) ( ) ( ) ( ) 3 ( ) ( ) c α φ φ φ φ Sice φ, φ, φ3,... φ ae liealy idepedet, by theoem.3.5, the detemiat of the coefficiets i.e. W( φ, φ, φ3,... φ)( ). Thus the coefficiet mati is ivetible. Theefoe thee is a uique set of costats c, c, c3,..., c satisfyig above system of equatios. Fo this choice of c, c, c3,..., c the fuctio φ( ) cφ( ) + cφ( ) + c3φ3( ) cφ( ) will be the desied solutio. Theoem.3.7 : Let φ, φ, φ3,... φ be liealy idepedet solutios of L(y) o a iteval I. If c, c, c3,..., c ae ay costats φ( ) cφ( ) + cφ( ) + c3φ3( ) cφ( ) is a solutio ad evey solutio may be epeseted i this fom. Poof : Sice φ i, i,, 3... is solutio of L(y), L( φ i), i,, Theefoe L( φ) cl( φ) + c L( φ) + c3l( φ3) c L( φ) ad Diffeetial Equatios φ c φ + c φ + c φ + + c φ is a solutio of L(f ) (7)

36 Let f be ay solutio of L(y) ad be i I. Suppose ( ) φ( ) α, φ ( ) α, φ ( ) α,..., φ ( ) α. Diffeetial Equatios 3 By eistece theoem.3.6 thee eist uique costats c, c, c3,..., c such that ψ cφ+ cφ + c3φ cφ is a solutio of L(y) o I satisfyig ( ) 3 ψ ( ) α, ψ ( ) α, ψ ( ) α,..., ψ ( ) α The uiqueess theoem.3.4 implies that f y. Thus φ cφ+ cφ + c3φ c φ. Theoem.3.8 Let φ, φ, φ3,... φ be solutios of L(y) o a iteval I costaiig a poit. The a ( ) 3 e 3 W(,,,... )( ) W(,,,... )( ) φ φ φ φ φ φ φ φ Poof : φ( ) φ( ) φ3( ) L φ( ) φ ( ) φ ( ) φ 3 ( ) L φ ( ) W( φ, φ, φ3,..., φ)( ) φ ( ) φ ( ) φ 3 ( ) L φ ( ) M M M M ( ) ( ) ( ) ( ) 3 φ ( ) φ ( ) φ ( ) φ ( ) By diffeetiatig above detemiat ow-wise we get, φ φ φ L φ 3 φ φ φ 3 L φ W ( φ, φ, φ3,..., φ)( ) φ φ φ 3 L φ M M M M ( ) ( ) ( ) ( ) 3 φ φ φ φ φ φ φ3 L φ φ φ φ3 L φ φ φ φ L φ φ φ φ L φ φ φ φ 3 L φ φ φ φ 3 L φ M M M M M M M M ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 L 3 L φ φ φ φ φ φ φ φ Sice two ows ae idetical the value of fist ( ) detemiats is zeo. Theefoe φ φ φ3 L φ φ φ φ L φ 3 W ( φ, φ, φ3,..., φ)( ) φ φ φ 3 L φ M M M M ( ) ( ) ( ) ( ) 3 φ φ φ φ (8)

37 i i i Sice each φ i, i,,3,..., is a solutio of L(y) ( ) ( ) ( ) φ ( a φ + a φ ( 3) i aφi + a3 φ... + ). Hece, W ( φ, φ, φ,..., φ )( ) 3 φ ( ) φ ( ) φ ( ) L φ ( ) 3 φ ( ) φ ( ) φ ( ) L φ ( ) 3 M M M L M ( ) ( ) ( ) 3 ( ) ( ) φ ( ) φ ( ) φ3 L ( ) φ φ ( ) φ ( ) φ ( ) φ ( ) a ( ) a ( ) a ( ) a ( ) Sice, φ ( ) φ ( ) φ ( ) L φ ( ) 3 φ ( ) φ ( ) φ ( ) L φ ( ) 3 M M M L M ( ) ( ) ( ) ( ) 3 ( ) ( ) ( ) ( ) φ φ φ3 L φ φ ( ) φ ( ) φ ( ) φ ( ) a ( ) a ( ) a ( ) a ( ) fo, 3, 4,..., as two ows of the detemiat ae costat multiplies of each othe ae Thus, φ( ) φ( ) φ3( ) L φ( ) φ ( ) φ ( ) φ 3 ( ) L φ ( ) W ( φ, φ, φ3,..., φ)( ) a φ ( ) φ ( ) φ 3 ( ) L φ ( ) M M M M ( ) ( ) ( ) ( ) 3 L φ ( ) φ ( ) φ ( ) φ ( ) a W( φ, φ, φ,..., φ )( ) 3 Thus W + aw. O itegatig this equatio betwee the limits to we get, a W( ) a W( ) e e a ( ) o W( ) e W( ) a ( ) 3 3 Thus W( φ, φ, φ,..., φ )( ) e W( φ, φ, φ,..., φ )( ) Theoem.3.9 Let φ, φ, φ3,... φ be solutios of L(y) o a iteval I cotaiig. The they ae liealy idepedet o I if ad oly if W( φ, φ, φ3,..., φ)( ) Diffeetial Equatios (9)

38 Poof : By theoem.3.5 the solutios φ, φ, φ3,... φ of L(y) ae liealy idepedet o a iteval I if ad oly if W( φ, φ, φ3,..., φ)( ) fo all i I. a ( ) But W( φ, φ, φ3,..., φ)( ) e W( φ, φ, φ3,..., φ)( ) (by theoem.3.8.) Theefoe W( φ, φ, φ3,..., φ)( ) if ad oly if W( φ, φ, φ3,..., φ)( ) ad the esult follows. Q.. Coside the equatio (5) (4) EXAMPLES y y y + y (a) Compute five liealy idepedet solutios. (b)compute the woia of the solutios foud i (a). (c) Fid that solutio f satisfyig (4) φ(), φ () φ () φ () φ (). As (a) : The chaacteistic equatio 5 4 p () + 4 ( ) ( ) 4 ( )( ) ( )( + )( ) ( + ) ( ) ( + ) ( ) Thus the chaacteistic oots ae,,, i, i Theefoe φ( ) e, φ( ) e, φ3( ) e, φ4( ) si φ 5 ( ) cos ae solutios of the give diffeetial equatio. As (b) : a ( ) e W(,,,, )( ) W(,,,, )( ) φ φ φ φ φ φ φ φ φ φ Fo the give equatio a. Let the e W( φ, φ, φ, φ, φ )( ) W( φ, φ, φ, φ, φ )(). e e e si cos e ( + ) e e cos si e e e W( φ, φ, φ3, φ4, φ5)( ) ( + ) si cos e (3 + ) e e cos si e (4 + ) e e si cos Diffeetial Equatios (3)

39 W( φ, φ, φ3, φ4, φ5)() 3 4 The ow tasfomatios R R, R3 R, R4 R, R5 R gives W( φ, φ, φ3, φ4, φ 5)() As (c) : Thus, φ φ φ3 φ4 φ5 e φ φ φ3 φ4 φ5 W(,,,, ) W(,,,, )() 3e The geeal solutio f is φ ( ) c e + c e + c e + c si+ c cos The iitial coditios system of equatios (iv) φ(), φ () φ () φ () φ () gives the followig + c c c3 3 c4 4 c 5 The ow tasfomatio R R, R3 R, R4 R, R5 Rgives Diffeetial Equatios (3)

40 + c c c3 3 c4 4 c 5 Solvig the above system of equatios simultaeously we get the values of c, c, c 3, c 4, c 5. Fom last equatio we get 4c gives c 4 Fom the thid ow of the above system we get, c c5 gives c5 4 Fom secod ad fouth ow we get, c c + c c c c3 c4 c5 Substitutio of c ad c 5 i above equatios give c3+ c4 c3 c4 Thus, c3, c Fom fist ow we get, c 8 Thus, φ ( ) ce + ce + c3e + c4si+ c5cos 5 e e + e si+ cos is the equied solutio. Q.. Fid all solutios of the followig equatios. As. (a) : (a) y 8y (b) (4) y + 6y (c) y 5y + 6y (d) ( iv y ) 6y (e) y 3 y y (f) (4) y + 5y + 4y The chaacteistic polyomial is Thus, thee liealy idepedet solutios ae give by Diffeetial Equatios 3 p () 8ad its oots ae, + 3 i, 3i i i ( 3) ( 3), + i i e e, e ad ay solutio f has the fom φ( ) ce ( + 3 ) + ce ( 3 ) + c3e whee c, c, c 3 ae ay costats. (3)

41 As. (b) : The chaacteistic polyomial is 4 p () + 6 Diffeetial Equatios 4 4 p( ) ( i) ( + ( i) ) ( ( i) ) ( i ( i) ) ( ( i) ) ( + i i)( i i)( + i)( i) Thus, p( ) ( + i i)( i i)( + i)( i) \ π π i cos + isi e i π π π i i 4 π π i e e cos + isi i i( + i) + i Theefoe i, i i The oots of chaacteistic polyomial ae ( + i), ( + i), ( + i), ( + i) Thus fou liealy idepedet solutios ae ad evey solutio f has the fom ( i ) ( + i) ( + i ) ( i ) e, e, e, ( i ) ( + i ) ( + i ) ( i ) 3 4 φ( ) ce + c e + c e + c e As. (c) : The chaacteistic polyomial is p () 5 + 6ad its oots ae, 3,. Thus thee liealy idepedet solutios ae give by, e 3, e ad ay solutio f has the fom 3 φ ( ) c e + c e + c 3 e 3 As. (d) : The chaacteistic polyomial is 4 p ( ) 6 ( + 4)( 4) ( + i)( i) ( + )( ) ad its oots ae,, i, i. Thus fou liealy idepedet solutios ae give by e, e, cos, si ad evey solutio f has the fom 3 4 φ ( ) c e + c e + c cos+ c si As. (e) : The chaacteistic polyomial is 3 p ( ) 3 ( + ))( ) ad its oots ae,,. (+ 5 ) 5 Thus, thee liealy idepedet solutios ae e, e, e, ad evey solutio f has the fom + 5 ( ) 3 φ( ) c e + c e + c e 5 ( ) (33)

42 As. (f) : The chaacteistic polyomial is 4 p () ( + 4)( + ) ad its oots ae i, i, i, i. Thus fou liealy idepedet solutios ae cos, si, cos, si ad evey solutio f has the fom φ ( ) c cos, + c si+ c cos+ c si. 3 4 Q.3. Coside the equatio y 4y (a) Compute thee liealy idepedet solutios. (b)compute the woia of the solutios foud i (a). (c) Fid the solutio f satisfyig φ(), φ (), φ () As. (a) : The chaacteistic polyomial p () 4ad its oots ae,,. Thus, thee liealy idepedet solutio ae e, e, e ad evey solutio f has the fom 3 As. (b) : 3 φ ( ) c + c e + c e ( ) 3 e 3 W( φ, φ, φ )( ) W( φ, φ, φ )() 3 W( φ, φ, φ )( ) e e e e 4e 4e 3 W( φ, φ, φ )() 4 4 Thus, W( φ, φ, φ 3)( ) 6. As. (c) : φ(), φ (), φ (), R 3 R gives 3 φ 3 φ( ) c + c e + c e, () c + c + c ad so o c c 4 4 c3 c c 8 c3 Diffeetial Equatios (34)

43 Theefoe c3, c c3 c c3 c 4 4 c + c + c c 3 Thus, ( ) c ce c3e ( e e ) φ + + is the equied solutio. 4 EXERCISE. Ae the followig statemets tue o false? (a) If φ, φ, φ3,..., φ ae liealy idepedet fuctios o a iteval I, the ay subset of them foms a liealy idepedet set of fuctios o I. (b) If φ, φ, φ3,..., φ ae liealy depedet fuctios o a iteval I, the ay subset of them foms a liealy depedet set of fuctios o I.. Ae the followig sets of fuctios defied o < < liealy idepedet o depedet? why? (a) φ ( ), φ ( ), φ ( ) 3 (b) i φ φ φ3 ( ) e, ( ) si, ( ) cos (c) φ ( ), φ ( ) e, φ ( ) 3 3. Fid a basis of solutios of the diffeetial equatios. (a) y + 5y + 4 (b) y + 6y + y + 8y (c) y (4) y 4. Fid the geeal solutio of each of the followig equatios. (i) 6 y y + 4y (As. y( ) c e + c e ) 4 3 (ii) ( + ) ( ) y + y y y ce + ce (As. ( ) ) (iii) 3 y + y 6y (As. y( ) c + c e + c e ) 3 (iv) (4) y y y c+ c+ c3e + c4e (As. ( ) ) (v) y + 8y ( As. y( ) ce + ce + c3e ) 5. Fo each of the followig equatios fid a paticula solutio which satisfies the give iitial coditios. (i) y, y(), y () (ii) y + 4y + 4y, y(), y () Diffeetial Equatios (35)

44 (iii) y y + 5y, y(), y () 4 (iv) y y y y( π ) y ( π ) 4 +,, (v) 3y + 5 y + y y, y(), y (), y () [As. : (i) y( ) 3, (ii) y( ) (+ 3 ) e (iii) y( ) e (cos+ si ) (iv) e 4 π si 4 (v) y e + e.] As. : (a) Tue (b) false As. : (a) idepedet (b) depedet (iii) idepedet As. 3 : (a) 4 φ( ) e, φ( ) e (b) φ( ) e, φ( ) e, φ3( ) e 3 4 (c) φ ( ) e, φ ( ) e, φ ( ) cos, φ ( ) si Uit 4 : The No-Homogeeous Equatio of Ode We ow etu to the th ode o-homogeeous liea diffeetial equatio with costat coefficiets. I the fist pat we will discuss the method of fidig all solutios of the secod ode o-homogeeous equatio. Ly ( ) y + ay + ay b ( ), Whee b is some cotiuous fuctio o a iteval I. The geeal solutio of the above equatio is y ( ) y( ) + y ( ), c p whee, y c (), the complemetay fuctio is the geeal solutio of the elated homogeous equatio ad y p () is a paticula solutio of the equatio. Suppose we ow that y p is a paticula solutio of the equatio L(y) b() ad let y be ay othe solutio. The, L( ψ ψ ) L( ψ) L( ψ ) b( ) b( ) p p o I. This shows that y y p is a solutio of the homogeous equatio L(y). Theefoe if f, f ae liealy idepedet solutios of L(y), thee ae uique costats c, c such that Diffeetial Equatios (36)

45 ψ ψ p c φ + c φ I othe wods evey solutio y of L(y) b () ca be witte i the fom ψ ψ p + cφ+ cφ The poblem of fidig all solutios of L(y) b () educes to fidig a paticula solutio y p. Theoem.4. Let b() be cotiuous o a iteval I. Evey solutio y of L(y) b () o I ca be witte as ψ ψ p + cφ+ cφ. Whee y p is a paticula solutio, f, f ae two liealy idepedet solutios of L(y) ad c, c ae costats. A paticula solutio y p is give by [ φ( t) φ( ) φ( ) φ( t)] b( t) ψ p( ) dt. W( φ, φ )( t) Covesely evey such y is a solutios of L(y) b () Poof : Let y ad y p be two solutios of Ly ( ) y + ay + ay b The L( ψ ψ ) L( ψ) L( ψ ) p p This shows that ψ ψ p is a solutio of a homogeeous equatio L(y). By theoem.. thee eist two liealy idepedet solutios f, f ad evey solutio of L(y) is of the fom cφ+ cφwhee c ad c ae costats. Such a fuctio cφ+ cφcaot be a solutio of L(y) b() uless b() o I. Suppose φ( ) u( ) φ( ) + u( ) φ( ) is a solutio of L(y) b() o I. (This pocedue is called as the vaiatio of costats.) The ( u φ + u φ ) + a ( u φ + u φ ) + a ( u φ + u φ ) b( ) i.e. a( uφ+ uφ) + a( u φ+ uφ + u φ + uφ ) Theefoe + ( u φ + u φ + u φ + u φ + u φ + u φ ) b( ) u ( φ + a φ + a φ ) + u ( φ + a φ + a φ ) + ( φ u + φ u ) + ( φ u + φ u ) + a ( φ u + φ u ) b( ) i.e. φ + φ + φ + φ + φ + φ ( u u ) ( u u ) a ( u u ) b( ) Diffeetial Equatios (37)

46 Obseve that if φ u + φ u ( ) ( ) ( ) the φu + φu φu + φu + φu + φu ad φ u + φ u b( ) Thus if we ca fid two fuctios u () ad u () such that φ The u φ uφ u + φu φ u + φ u b( ) + will satisfy L(y) b(). ad O solvig above two equatios fo u ad u we get, φb ( ), φ b u u ( ), W( φ, φ) W( φ, φ) Itegatio of above equatio betwee the limits to povides φ() t b() t u( ) dt+ u( ) W( φ, φ )( t) φ() t b() t + W( φ, φ)( t) u ( ) dt u ( ). The solutio u φ + uφtaes the fom φ() t b() t φ( ) φ( ) dt+ u( ) W( φ, φ)( t) φ() t b() t + φ( ) + dt+ u( ) W( φ, φ)( ) t The tem φ ( u ) ( ) + φ ( u ) ( ) is a complemetay fuctio o the solutio of coespodig homogeeous equatio L(y) ad the paticula solutio taes the fom φ t b t φ φ φ t φ φ () () () t b() t ψ p( ) φ ( ) dt+ φ ( ) dt W(, )( ) W(, )( t) [ φ( t) φ( ) φ( t) φ( )] b( t) ψ p( ) dt W( φ, φ )( t) The fuctio y p () is a solutio of L(y) b (). Theoem.4. povides a method to fid a solutio of secod ode o-homogeeous diffeetial equatio with costat coefficiets. The same pocedue ca be geealized fo the o-homogeeous equatio of ode. Diffeetial Equatios (38)

47 Theoem.4. Let b be cotiuous o a iteval I ad let φ, φ, φ 3,..., φ be liealy idepedet solutios of ( ) ( ) ( ) Ly ( ) y + ay + ay ay o I. Evey solutio y of L(y) b() ca be witte as ψ ψ + cφ + c φ + c φ + + c φ p Whee y p is a paticula solutio of L(y) b() ad c, c, c3,..., c ae costats. Evey such y is a solutio of L(y) b(). A paticula solutio y p is give by W φ 3 () t b() t ψ p( ) ( ) dt. W( φ, φ, φ,..., φ )( t) Poof : The poof is simila to the poof of theoem.4. Let b be cotiuous fuctio o a iteval I. Coside the diffeetial equatio ( ) ( ) ( ) Ly ( ) y + ay + ay ay b ( ) whee, a, a, a 3,..., a ae costats. If y p is a paticula solutio of L(y) b() ad y is ay othe solutio of L(y) b(), the L( ψ ψ ) L( ψ) L( ψ ) b( ) b( ) p p ad y y p is a solutio of coespodig homogeeous equatio L(y). (is called subtactio piciple). Thus ay solutio y of L(y) b() ca be witte i the fom ψ ψ + φ + φ + φ + + φ p c c c c whee, y p is a paticula solutio of L(y) b(), the fuctios φ, φ, φ 3,..., φ ae liealy idepedet solutios of L(y) (detemied i theoem.3.) ad c, c, c3,..., c ae costats. To fid a paticula solutio y p we use the vaiatio of costats method. Suppose ψ u ( ) φ ( ) + u ( ) φ ( ) + u ( ) φ ( ) u ( ) φ ( ) p 3 3 is a solutio of L(y) b(). Sice y p is a solutio it satisfies the equatio i.e L(y p ) b(). ψ uφ + u φ + u φ + + u φ p uiφi i The, ψ u φ + u φ+ uφ + uφ u φ + u φ p ( u φ + u φ + u φ u φ ) + ( u φ + u φ u φ ) 3 3 u φ i i ui φi i i + Let u φ the ψ u φ i i p i i We have ψ u φ + u φ p i i i i Diffeetial Equatios (39)